Traffic flow instabilities include slowdowns, back-ups, and alternating waves of stop-and-go driving, which often appear to occur for no obvious reason such as an accident or road construction work. These periodic and often rapid variations in speed and traffic density reduce overall traffic throughput on highways and increase the danger of collisions, cause higher fuel consumption and wear and tear on vehicles, faster abrasion of roadways, and waste the time and fray the nerves of motorists.
Traffic flow instabilities are best understood in light of the relationship between vehicle density and speed. Vehicle density is the number of vehicles in a unit length of a roadway and depends on the average length of such vehicles and the distance between vehicles. Ideally, drivers will attempt to keep a safe distance from vehicles that they are following so that if a lead vehicle were to suddenly apply its brakes, the trailing vehicle could apply its brakes in time to avoid a collision. The safe distance depends on speed and reaction time, specificallysmin=T×v where smin is the minimum safe distance, T is reaction time, and v is speed. In practice one factors in a margin of safety by using a value for T that is considerably larger than the typical reaction time. A well known rule of thumb for maintaining a safe minimum distance, for example, calls for a driver to allow one car length between his vehicle and a leading vehicle for each 10 miles per hour of speed. Hence, a motorist travelling at 60 miles per hour should allow a gap of six car lengths between his vehicle and the vehicle he is following. But, in any case, what is important here is that the minimum safe distance is proportional to velocity or speed.
In an ideal world, where drivers maintain a safe minimum distance between their vehicles and the vehicles they are following during rush hours or other peak driving periods, traffic density should decrease at higher speeds as vehicle spacing increases to maintain a safe minimum distance. Conversely, traffic density will increase at lower speeds as the safe minimum distance decreases at lower speeds. Thus, if all drivers maintain a safe minimum distance from vehicles ahead, traffic density (vehicles per unit distance) is inversely proportional to the sum of vehicle length and safe minimum distance, which may be expressed as:ρ=1/(d+T×v)where ρ equals traffic density (vehicles per unit distance) and d is the length of a vehicle. Vehicle throughput (vehicles passing a fixed point per unit time) equals density times speed, sor=v/(d+T×v)where r is the throughput (vehicles per second). Thus, at low speeds, throughput is approximately proportional to speed orr≈v/d (for v<d/T).
However, even though throughput increases with speed, it does so more and more slowly as speed increases, and throughput approaches an asymptotic value ofrmax=1/T when v becomes large. If one conservatively assumed total reaction time to be one second, for example, throughput would be ultimately limited to one vehicle per second (or 3600 vehicles per hour per lane).
Unfortunately, in the real world, traffic throughput is not steady during peak driving periods, in part because many drivers do not consistently maintain safe minimum distances. For example, some drivers tailgate or switch lanes precipitously and then reduce their speed to avoid a collision or to establish a safe minimum distance; other drivers attempt to maintain a minimum distance that is less than safe; finally, other drivers maintain a minimum distance that is longer or greater than optimum and cause trailing vehicles to reduce the gap between vehicles. In all of these events, when vehicle density is above a certain level, the application of the brakes of one vehicle to avoid a collision or to establish a safer minimum distance will cause a cascade effect as each following driver applies his or her brakes to compensate for the reduction in the speed of the leading vehicle. Thus, a wave travels backward through the traffic, with amplitude increasing with distance from the original disturbance.
These waves or instabilities increase traffic density because lower speed causes higher densities, and higher densities cause even lower speeds. Indeed, because of this positive feedback, waves in density and speed grow in amplitude until the speed at the low point of the cycle drops to zero and traffic is brought to a standstill. The overall throughput in the presence of these wavelike disturbances of traffic flow is much lower than would be possible with steady flow in part because the average speed is much lower.
Another way of understanding the source of traffic flow instabilities is to consider each driver and vehicle combination as a control system (or a controlled vehicle) that adjusts speed in response to the relative position of a vehicle ahead or leading vehicle, as well as the relative speed of the leading vehicle. Each driver and vehicle can be thought of as a system with an input (the relative position and speed of the leading vehicle) and an output (acceleration or deceleration, and hence, indirectly, speed and position of the controlled vehicle itself). Such a control system can be said to have a “gain” which is the ratio of amplitude of the output to that of the input. If, for example, the control system is able to accurately follow the input, then it has gain of one. It is well known in the art of control systems that, if there exists any motion waveform that is amplified with a gain of more than one by the control system, even if only by a small amount, then cascading many such control systems leads to increasing amplitudes of deviation from the average the further back one goes from the initial disturbance.
So, if there is a frequency of oscillation for which the amplitude of the oscillations at the output of a control system is larger than it is at its input, there will be a problem when many such systems are cascaded. That is, if the systems have gain greater than one for waves of some frequency, then, when multiplied together, these gains produce larger and larger overall gain, as more and more systems are cascaded. For stability, the gains need to be strictly less than one for all frequencies. In other words, the amplitude of the response to a disturbance needs to be less than the amplitude of the disturbance itself.
Yet, vehicles also cannot successfully avoid collisions when there are large amplitude oscillations, unless the gain is higher than one at some frequencies. For example, when a leading vehicle periodically speeds up and slows down it will alternately be ahead of, and then behind, where it would have been if it had moved with a steady velocity equal to its average velocity. The maximum departure from the average position is called the amplitude of the oscillation.
If the gain of the control system of a vehicle following a controlled vehicle is one, then the following vehicle will reproduce exactly the same increases and decreases in velocity (ignoring delay in the control system for the moment) and hence the same departures from the average position. In this case, the separation between the two vehicles is constant and no collision can occur.
However, if the gain of the control system of the following vehicle is less than one, the following vehicle, while still reproducing the motion of the leading vehicle, will do so with reduced amplitude. The two vehicles will collide if the difference in amplitude between their motions exceeds their initial separation. This is most easily seen when the gain is zero; that is, when the following vehicle moves at fixed speed. In that case, as the amplitude of the oscillation of the leading vehicle is increased, a point is reached where the leading vehicle lags so far behind its average position that it drops back to where the following vehicle currently is. When the gain is non-zero, the following vehicle's oscillation will tend to reduce the chance of collision for a fixed amplitude of oscillation, but there will still be some amplitude for which collision is unavoidable.
Overall then, it appears that the control system gain cannot be greater than one, and yet cannot be less than one for safe operation under all conditions. Thus, the problem of traffic flow instability is simply unavoidable when the driver and vehicle are modeled as a simple “car following” control system.
Further, it is also well known in the art of control systems that any delay in a feedback loop can lead to instabilities. Thus the finite reaction time of a driver (and the dynamics of the vehicle and its control system) plays a role in producing instabilities. Shorter reaction times allow higher throughput, because they allow the separation between vehicles to be smaller, but for any given reaction time there will be a critical density above which perturbations are amplified and will propagate. Further, the components of a driver/vehicle control system model are non-linear because speed cannot become negative or exceed some upper limit, and the distance between vehicles cannot become negative either. These non-linearities, along with positive feedback, create the classic conditions for instabilities or even chaotic behavior.
The above is but one way of understanding the origins of traffic flow instabilities. Many different models have been made of traffic flow using mathematical tools such as differential equations, difference equations, cellular automata, fluid flow models, particle tracking, and so-called “car following” models. All show travelling waves of instabilities and amplification of these waves above some critical density. None, however, suggest a solution to the problem.
Because the incidence of large density and speed fluctuations increase with traffic flow density, one approach to the problem is to reduce traffic density by building more roads or more lanes per road. More road construction would certainly help reduce density, but it is not a viable option in many cases in view of land use restrictions or inadequate financing.
Limiting or “metering” roadway access at entry points at or below some target value also certainly helps to reduce traffic density, but it forces roadways to operate well below their maximum carrying capacity.
Another approach would be to reduce driver reaction time to allow vehicles to follow each other more closely at higher speeds without danger of collision by eliminating the standard arrangement of accelerator and brake pedal, which unnecessarily lengthens reaction time because the foot has to be lifted from one and applied to the other. It is unlikely, however, that the standard brake and gas pedal design will ever be replaced.
An automated control system, somewhat analogous to cruise control, using automatic feedback based on sensor readings can reduce “reaction time.” But, as pointed out above, there is still a critical density above which instabilities occur. In addition, vehicles with such automated control may cause further instabilities when mixed with vehicles controlled by drivers, because the automated vehicles—with faster reaction times—will appear to be “tail gating” at uncomfortably close range.
Fully automated control systems with high speed communication between vehicles can allow a lead vehicle to directly control several following vehicles that travel together in “platoons” much as the engine of a train controls the motion of attached carriages. However, such an approach is best suited for a separate road system limited to platoons of fully automated vehicles. It is unlikely to be safe in a mixed environment with some vehicles controlled by human drivers. Thus, this approach would either require duplicating existing infrastructure or forcing all vehicles to be converted to completely automatic operation. There are also complex issues concerning the formation of platoons, and how they would enter and exit a highway, or change lanes.
Suggestions for automatic “distance keeping” by a following vehicle with reference to a leading vehicle date at least to the work of Dr. Ichiro Masaki (U.S. Pat. No. 4,987,357). His automobile cruise-control system uses machine vision technology to automatically adjust a controlled vehicle's speed to keep a safe distance from a leading vehicle.
All such suggestions for “adaptive cruise control” rely on information about a leading vehicle, typically the distance to that vehicle and the difference in speed between the leading and the controlled vehicle. A traditional “car following” control system is illustrated in FIG. 1 where the controlled vehicle ‘C’ takes as input the distance to the leading vehicle ‘L’ (d1), and the relative speed of the leading vehicle (v1−vc). A car following system may also have additional inputs, such as the speed of the controlled vehicle vc itself, as well as parameters that control the operation of the controlled vehicle, such as the desired speed vdes and a maximum allowed speed vmax as depicted in FIG. 2a. As shown in FIG. 2a, a car following control system outputs a positive acceleration command a, or a negative acceleration command a, to the drive control system in order to speed up or slow down respectively. The control system could alternatively, for example, output a speed set point for the drive control system, or activate mechanical brakes and/or a regenerative braking system.
Importantly, the information flow in a “car following” system is strictly one-way: from leading to controlled vehicle, or in other words from front to back. The controlled vehicle's speed is adjusted based on information of the relative position and speed of the vehicle in front of it. Only the leading vehicle influences what is behind it. There is no information propagating forward from the following vehicle.